
Discrete Random Variable
I can't seem to figure out the second part of this question.
Herman plays chess with a friend. In the chess games, Herman wins 30% of the time and loses 20% of the time. Herman pays his friend $2 every time he loses and collects $1 from his friend every time he wins. The remainder of the games end in a draw and no money changes hands. Find the expected value and the standard deviation of Herman's net gain after two chess games.
For the first part, what I did was 1(0.3) + (2)(0.2) + 0(0.5) = 0.1 x 2 (two games) = 0.2
The back of the book shows the same answer but I'm not sure if my method is correct. Is it appropriate to multiply the answer by 2 to get 0.2 or was I suppose to double the money values (1, 2, and 0)? Both yield the same answer.
And as for the standard deviation, I'm getting an answer close to but off of 1.476 (answer from the back) which makes me think I'm doing it wrong. Can someone please show me how to go through this question?
Thanks

I get
$\displaystyle \displaystyle SD(X) = \sqrt{E(X^2)[E(X)]^2} = \sqrt{1^2\times 0.3+(2)^2\times 0.2(0.1)^2} = 1.044$